In a recent work [1]}, W. Zhou and the second named author proved the following statement: any compact Hermitian threefold with vanishing real bisectional curvature must be Chern flat. This is a special case of balanced threefolds. Recall that real bisectional curvature is a curvature notion on Hermitian manifolds introduced by X. Yang and the second named author in [2]}. It is equivalent to \(H\)  in strength when the metric is Kähler, but slightly stronger than \(H\)  in the non-Kähler case, so the main result of [1]} is weaker than the balanced case of Conjecture 1 for \(n=3\)  and \(c=0\) .
